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Boolean random functions
Author(s) -
Serra J.
Publication year - 1989
Publication title -
journal of microscopy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.569
H-Index - 111
eISSN - 1365-2818
pISSN - 0022-2720
DOI - 10.1111/j.1365-2818.1989.tb02905.x
Subject(s) - boolean function , mathematics , boolean model , boolean expression , bounded function , discrete mathematics , function (biology) , infimum and supremum , set (abstract data type) , combinatorics , parity function , boolean network , computer science , mathematical analysis , evolutionary biology , biology , programming language
SUMMARY A Boolean function f in R n is the supremum of upper semi‐continuous random functions f'i which are almost surely positive, bounded with compact support and centred at the Poisson points ( i ). They generalize to functions of classical Boolean model for sets. The Boolean function f may be studied via its subgraph, i.e. as a random set in R n x R . The key notion is then the functional Q(Bt) , i.e. the probability that a compact set Bt centred at altitude t misses the subgraph of f . The general expression of Q(Bt) is given, and followed by a series of important derivations (volumes, gradients, numbers of summits, etc). Theorems of structure are given: they concern the properties of infinite divisibility for the sup, and domains of attraction for Boolean functions. The last sections are devoted to the study of two particular Boolean functions; emphasis is put on the stereological implications of the approach. A critical example illustrates the theory.